Introduction Difference-in-Differences Multidimensional RD Control Variables Multidimensional Regression Discontinuity and Regression Kink Designs with Difference-in-Differences Rafael P. Ribas University of Amsterdam Stata Conference Chicago, July 28, 2016
Introduction Difference-in-Differences Multidimensional RD Control Variables Motivation Regression Discontinuity (RD) designs have been broadly applied. However, non-parametric estimation is restricted to simple specifications. I.e., cross-sectional data with one running variable. Thus some papers still use parametric polynomial forms and/or arbitrary bandwidths. For instance, Dell (2010, Econometrica ) estimates a two-dimensional RD. Grembi et al. (2016, AEJ:AE ) estimates Difference-in-Discontinuities. The goal is to create a program (such as rdrobust ) that accommodates more flexible specifications. Ribas, UvA Regression Discontinuity 1 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Motivation Regression Discontinuity (RD) designs have been broadly applied. However, non-parametric estimation is restricted to simple specifications. I.e., cross-sectional data with one running variable. Thus some papers still use parametric polynomial forms and/or arbitrary bandwidths. For instance, Dell (2010, Econometrica ) estimates a two-dimensional RD. Grembi et al. (2016, AEJ:AE ) estimates Difference-in-Discontinuities. The goal is to create a program (such as rdrobust ) that accommodates more flexible specifications. Ribas, UvA Regression Discontinuity 1 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Overview The ddrd package is built upon rdrobust package, including the following options: 1 Difference-in-Discontinuities (DiD) and Difference-in-Kinks (DiK) 2 Multiple running variables 3 Analytic weights ( aweight ) 4 Control variables 5 Heterogeneous effect through linear interaction (in progress). All options are taken into account when computing the optimal bandwidth, using ddbwsel . The estimator changes, so does the procedure. Ribas, UvA Regression Discontinuity 2 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Difference-in-Discontinuity/Kink, Notation Let µ t ( x ) = E [ Y | X = x, t ] and µ ( v ) t ( x ) = ∂ v E [ Y | X = x,t ] . ( ∂x ) v Then the conventional sharp RD/RK estimand is: x → 0 + µ ( v ) x → 0 − µ ( v ) t ( x ) = µ ( v ) t + − µ ( v ) τ v,t = lim t ( x ) − lim t − The DiD/DiK estimand is: � � ∆ τ v = µ ( v ) 1+ − µ ( v ) µ ( v ) 0+ − µ ( v ) 1 − − 0 − Ribas, UvA Regression Discontinuity 3 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Optimal Bandwidth, h ∗ Two methods based on the mean square error (MSE): � 1 � C ( K ) Var(ˆ τ v ) 5 n − 1 h ∗ MSE = 5 τ v ) 2 Bias(ˆ Imbens and Kalyanaraman (2012), IK. Calonico, Cattaneo and Titiunik (2014), CCT. They differ in the way Var(ˆ τ v ) and Bias(ˆ τ v ) are estimated. For DiD/DiK, the trick is to replace ˆ τ v by ∆ˆ τ v . That’s what ddbwsel does. While ddrd calculates the robust, bias-corrected confidence intervals for ∆ˆ τ v , as proposed by CCT. Ribas, UvA Regression Discontinuity 4 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Optimal Bandwidth, h ∗ Two methods based on the mean square error (MSE): � 1 � C ( K ) Var(ˆ τ v ) 5 n − 1 h ∗ MSE = 5 τ v ) 2 Bias(ˆ Imbens and Kalyanaraman (2012), IK. Calonico, Cattaneo and Titiunik (2014), CCT. They differ in the way Var(ˆ τ v ) and Bias(ˆ τ v ) are estimated. For DiD/DiK, the trick is to replace ˆ τ v by ∆ˆ τ v . That’s what ddbwsel does. While ddrd calculates the robust, bias-corrected confidence intervals for ∆ˆ τ v , as proposed by CCT. Ribas, UvA Regression Discontinuity 4 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Application: Retirement and Payroll Credit in Brazil In 2003, Brazil passed a legislation regulating payroll lending. Loans for which interests are deducted from payroll check (Coelho et al., 2012). It represented a “kink” in loans to pensioners. Ribas, UvA Regression Discontinuity 5 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Application: Retirement and Payroll Credit in Brazil In 2003, Brazil passed a legislation regulating payroll lending. Loans for which interests are deducted from payroll check (Coelho et al., 2012). It represented a “kink” in loans to pensioners. Before, 2002 After, 2008 .3 .3 .2 .2 borrower borrower .1 .1 0 0 30 40 50 60 70 80 90 30 40 50 60 70 80 90 age age Ribas, UvA Regression Discontinuity 5 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Application: Retirement and Payroll Credit in Brazil Optimal bandwidth for Difference-in-Kink at age 60: . ddbwsel borrower aged [aw=weight], time(time) c(60) deriv(1) all Computing CCT bandwidth selector. Computing IK bandwidth selector. Bandwidth estimators for local polynomial regression Cutoff c = 60 | Left of c Right of c Number of obs = 53757 ----------------------+---------------------- NN matches = 3 Number of obs, t = 0 | 20836 4484 Kernel type = Triangular Number of obs, t = 1 | 22609 5828 Order loc. poly. (p) | 2 2 Order bias (q) | 3 3 Range of aged, t = 0 | 29.996 29.999 Range of aged, t = 1 | 29.996 29.996 ---------------------------------------------- Method | h b rho ----------+----------------------------------- CCT | 12.45718 18.73484 .6649206 IK | 14.46675 11.01818 1.312989 ---------------------------------------------- Ribas, UvA Regression Discontinuity 6 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Application: Retirement and Payroll Credit in Brazil ddrd output: . ddrd borrower aged [aw=weight], time(time) c(60) deriv(1) b(‘b’) h(‘h’) Preparing data. Calculating predicted outcome per sample. Estimation completed. Estimates using local polynomial regression. Derivative of order 1. Cutoff c = 60 | Left of c Right of c Number of obs = 27093 ----------------------+---------------------- NN matches = 3 Number of obs, t = 0 | 6117 3081 BW type = Manual Number of obs, t = 1 | 7319 4001 Kernel type = Triangular Order loc. poly. (p) | 2 2 Order bias (q) | 3 3 BW loc. poly. (h) | 12.457 12.457 BW bias (b) | 18.735 18.735 rho (h/b) | 0.665 0.665 Outcome: borrower. Running Variable: aged. -------------------------------------------------------------------------------------- Method | Coef. Std. Err. z P>|z| [95% Conf. Interval] ----------------------+--------------------------------------------------------------- Conventional | .0229 .0221 1.0362 0.300 -.020417 .066218 Robust | .0271 .03123 0.8680 0.385 -.034098 .088303 -------------------------------------------------------------------------------------- Ribas, UvA Regression Discontinuity 7 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Difference-in-Kink What if there is no cutoff and aged is a continuous treatment? Shift in level represents the first difference, while change in the slope represents the second difference. Difference-in-Difference with continuous treatment. Ribas, UvA Regression Discontinuity 8 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Difference-in-Kink Estimating changes in the first derivative at any part of the function: . ddrd borrower aged [aw=weight], time(time) c(60) deriv(1) nocut Preparing data. Computing bandwidth selectors. Calculating predicted outcome per sample. Estimation completed. Estimates using local polynomial regression. Derivative of order 1. Reference c = 60 | Time 0 Time 1 Number of obs = 53757 ----------------------+---------------------- NN matches = 3 Number of obs | 8433 10395 BW type = CCT Order loc. poly. (p) | 2 2 Kernel type = Triangular Order bias (q) | 3 3 BW loc. poly. (h) | 11.489 11.489 BW bias (b) | 16.813 16.813 rho (h/b) | 0.683 0.683 Outcome: borrower. Running Variable: aged. -------------------------------------------------------------------------------------- Method | Coef. Std. Err. z P>|z| [95% Conf. Interval] ----------------------+--------------------------------------------------------------- Conventional | .00473 .00161 2.9473 0.003 .001585 .007879 Robust | .00528 .0022 2.3988 0.016 .000966 .009598 -------------------------------------------------------------------------------------- Ribas, UvA Regression Discontinuity 9 / 16
Introduction Difference-in-Differences Multidimensional RD Control Variables Multidimensional RD, Notation Suppose X has k dimensions, i.e. X = { x 1 , · · · , x k } . Cutoff doesn’t have to be unique. Let c = { ( c 11 , · · · , c n 1 ) , · · · , ( c 1 L , · · · , c nL ) } be the cutoff hyperplane. It separates treated and control. z i indicates whether i is “intended for treatment” (in the treated set) or not (in the control set). Trick : pick one point in c , say c l = ( c 1 l , · · · , c nl ), and reduce X to one dimension by calculating the distance d ( x i , c l ) for every i . The new running variable is: r i = (2 · z i − 1) · d ( x i , c l ) . Ribas, UvA Regression Discontinuity 10 / 16
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